What is the Biblical basis for saying Lucifer is an angel?

You view is absolutely incorrect. There are more than 3 typs of beings. Look up the Cherubim. The bible talks about different beings.

The Hebrew term cherubim is cognate with the Assyrian term karabu, Akkadian term kuribu, and Babylonian term karabu; the Assyrian term means 'great, mighty', but the Akkadian and Babylonian cognates mean 'propitious, blessed'.[3][4] In some regions the Assyro-Babylonian term came to refer in particular to spirits which served the gods, in particular to the shedu (human-headed winged bulls);[4] the Assyrians sometimes referred to these as kirubu, a term grammatically related to karabu.[3] They were originally a version of the shedu, protective deities sometimes found as pairs of colossal statues either side of objects to be protected, such as doorways.[4][5] However, although the shedu were popular in Mesopotamia, archaeological remains from the Levant suggest that they were quite rare in the immediate vicinity of the Israelites.[5] The related Lammasu (human-headed winged lions — to which the sphinx is similar in appearance), on the other hand, were the most popular winged-creature in Phoenician art, and so scholars suspect that Cherubim were originally a form of Lammasu.[5] In particular, in a scene reminiscent of Ezekiel's dream, the Megiddo Ivories — ivory carvings found at Megiddo (which became a major Israelite city) — depict an unknown king being carried on his throne by hybrid winged-creatures.[6]

The bible NEVER gives the impression of angelic beings being the ONLY ones in existence besides man, god and angels. And in this instance Cherubims are not baby like beings, they are to be so big that a human can walk under them or even ride them as they fly.

Freds Bend put it the best way I have ever heard.

Also, angel may not be a species at all, but an office of position.

It is true it never talks about Satan BEING an angel.... BUT! The bible also states the "powers and the principalities"

Ephesians 6:12

For our struggle is not against flesh and blood, but against the rulers, against the authorities, against the powers of this dark world and against the spiritual forces of evil in the heavenly realms.

Evil can still be in the heavenly realms and FORCES is indeed plural...So I would take that and automatically think that angels are not the only things that turned evil. Correct?

Many commentators use the scripture in Ezekiel 28:11-17 to represent, symbolically, the angel called Lucifer or Satan who was involved with worship and was in the garden and was also among the fiery stones. (See Moses on the mountain for this reference.) All of the allusions in Ezekiel seem to say that the person/angel referred to was the devil/Satan/Lucifer. The most obvious interpretation of the scripture is that it is really about the King of Tyre but as in any scripture there is sometimes an immediate interpretation and a future completion.

If you look at the example in Isaiah 9:6 about Jesus and the reference to a "virgin birth," God speaks prophetically, "Behold a virgin shall bring forth a child." Most historians have said this refers to another event that happened within the life of the prophet; having said this, many of the gospel writers and early church writers have understood this as a direct prophetic word for the virgin birth of Jesus Christ. So the answer to the Question in "Where in the Bible does it refer to Lucifer?" could refer to the story of the King Tyre and the allusions to an angel in the garden of Eden and a perfectly created being that fell because of sin and worship of his own beauty in Ezekiel. There many allusions to Lucifer in Ezekiel in chapters 25, 28, 29, 35 and 39 as well.

"Where in the Bible does it say that Lucifer was one of the angels or an archangel?"

It doesn't.

“Lucifer” is actually a Hebrew word when translated means “shining one.” The Septuagint uses the Greek word that means “bringer of dawn.” That is why some translations render the original Hebrew “morning star” or “Daystar.” But Jerome’s Latin Vulgate uses “Lucifer” (light bearer), and this accounts for the appearance of that term in various versions of the Bible.

The 'name' Lucifer occurs once in the Scriptures (Isa. 14:12) and only in some versions of the Bible. Traditionally, Lucifer is a name that in English generally refers to the Devil before being cast from heaven. But this description is given to a man and not to a spirit creature as is further seen by the statement: “Down to Sheol you will be brought.” Sheol is the common grave of mankind—not a place occupied by Satan the Devil. Additionally, those seeing Lucifer brought into this condition ask: “Is this the MAN that was agitating the earth?” Clearly, “Lucifer” refers to a human, not to a spirit creature. - Isaiah 14:4, 15, 16.

Careful Exegesis has actually conclude that Isaiah did not have Satan in mind when he says "Lucifer," but actually the king of Babylon.

CONCERNING THE TRADITION OF LUCIFER BEING A NAME FOR THE DEVIL

I have read that before the rise of Christianity, it was the pseudepigrapha of Enochic Judaism who interpreted Isaiah 14:12-15 as applicable to Satan, and presented him as a fallen angel cast out of Heaven. Christian tradition, influenced by this presentation, came to use the Latin word for "morning star", lucifer, as a proper name ("Lucifer") for Satan as he was before his fall. As a result, "Lucifer has become a by-word for Satan in the Church and in popular literature", as in Dante Alighieri's Inferno and John Milton's Paradise Lost.

You may appreciate the following link to further information: SHINING ONE (Isaiah 14:12).